Normal-inverse Gaussian distribution

Normal-inverse Gaussian (NIG)

Parameters	$\mu$ location (real) $\alpha$ tail heaviness (real) $\beta$ asymmetry parameter (real) $\delta$ scale parameter (real) $\gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}$
Support	$x\in (-\infty ;+\infty )\!$
PDF	${\frac {\alpha \delta K_{1}\left(\alpha {\sqrt {\delta ^{2}+(x-\mu )^{2}}}\right)}{\pi {\sqrt {\delta ^{2}+(x-\mu )^{2}}}}}\;e^{\delta \gamma +\beta (x-\mu )}$ $K_{j}$ denotes a modified Bessel function of the second kind
Mean	$\mu +\delta \beta /\gamma$
Variance	$\delta \alpha ^{2}/\gamma ^{3}$
Skewness	$3\beta /(\alpha {\sqrt {\delta \gamma }})$
Excess kurtosis	$3(1+4\beta ^{2}/\alpha ^{2})/(\delta \gamma )$
MGF	$e^{\mu z+\delta (\gamma -{\sqrt {\alpha ^{2}-(\beta +z)^{2}}})}$
CF	$e^{i\mu z+\delta (\gamma -{\sqrt {\alpha ^{2}-(\beta +iz)^{2}}})}$

The normal-inverse Gaussian distribution (NIG, also known as the normal-Wald distribution) is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. The NIG distribution was noted by Blaesild in 1977 as a subclass of the generalised hyperbolic distribution discovered by Ole Barndorff-Nielsen. In the next year Barndorff-Nielsen published the NIG in another paper. It was introduced in the mathematical finance literature in 1997.

The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.

Properties

Moments

The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available.

Linear transformation

This class is closed under affine transformations, since it is a particular case of the Generalized hyperbolic distribution, which has the same property. If

$x\sim {\mathcal {NIG}}(\alpha ,\beta ,\delta ,\mu ){\text{ and }}y=ax+b,$

then

$y\sim {\mathcal {NIG}}{\bigl (}{\frac {\alpha }{\left|a\right|}},{\frac {\beta }{a}},\left|a\right|\delta ,a\mu +b{\bigr )}.$

Summation

This class is infinitely divisible, since it is a particular case of the Generalized hyperbolic distribution, which has the same property.

Convolution

The class of normal-inverse Gaussian distributions is closed under convolution in the following sense: if $X_{1}$ and $X_{2}$ are independent random variables that are NIG-distributed with the same values of the parameters $\alpha$ and $\beta$, but possibly different values of the location and scale parameters, $\mu _{1}$, $\delta _{1}$ and $\mu _{2},$ $\delta _{2}$, respectively, then $X_{1}+X_{2}$ is NIG-distributed with parameters $\alpha ,$ $\beta ,$$\mu _{1}+\mu _{2}$ and $\delta _{1}+\delta _{2}.$

Related distributions

The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the normal distribution, $N(\mu ,\sigma ^{2}),$ arises as a special case by setting $\beta =0,\delta =\sigma ^{2}\alpha ,$ and letting $\alpha \rightarrow \infty$.

Stochastic process

The normal-inverse Gaussian distribution can also be seen as the marginal distribution of the normal-inverse Gaussian process which provides an alternative way of explicitly constructing it. Starting with a drifting Brownian motion (Wiener process), $W^{(\gamma )}(t)=W(t)+\gamma t$, we can define the inverse Gaussian process $A_{t}=\inf\{s>0:W^{(\gamma )}(s)=\delta t\}.$ Then given a second independent drifting Brownian motion, $W^{(\beta )}(t)={\tilde {W}}(t)+\beta t$, the normal-inverse Gaussian process is the time-changed process $X_{t}=W^{(\beta )}(A_{t})$. The process $X(t)$ at time $t=1$ has the normal-inverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of Lévy processes.

As a variance-mean mixture

Let ${\mathcal {IG}}$ denote the inverse Gaussian distribution and ${\mathcal {N}}$ denote the normal distribution. Let $z\sim {\mathcal {IG}}(\delta ,\gamma )$, where $\gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}$; and let $x\sim {\mathcal {N}}(\mu +\beta z,z)$, then $x$ follows the NIG distribution, with parameters, $\alpha ,\beta ,\delta ,\mu$. This can be used to generate NIG variates by ancestral sampling. It can also be used to derive an EM algorithm for maximum-likelihood estimation of the NIG parameters.